Average Error: 43.9 → 11.4
Time: 18.7s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, a \cdot \left(4 \cdot c\right)\right) + \mathsf{fma}\left(b, b, -a \cdot \left(4 \cdot c\right)\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, a \cdot \left(4 \cdot c\right)\right) + \mathsf{fma}\left(b, b, -a \cdot \left(4 \cdot c\right)\right)} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1418911 = b;
        double r1418912 = -r1418911;
        double r1418913 = r1418911 * r1418911;
        double r1418914 = 4.0;
        double r1418915 = a;
        double r1418916 = r1418914 * r1418915;
        double r1418917 = c;
        double r1418918 = r1418916 * r1418917;
        double r1418919 = r1418913 - r1418918;
        double r1418920 = sqrt(r1418919);
        double r1418921 = r1418912 + r1418920;
        double r1418922 = 2.0;
        double r1418923 = r1418922 * r1418915;
        double r1418924 = r1418921 / r1418923;
        return r1418924;
}
↓
double f(double a, double b, double c) {
        double r1418925 = b;
        double r1418926 = 6.961131476357277e-08;
        bool r1418927 = r1418925 <= r1418926;
        double r1418928 = a;
        double r1418929 = -r1418928;
        double r1418930 = 4.0;
        double r1418931 = c;
        double r1418932 = r1418930 * r1418931;
        double r1418933 = r1418928 * r1418932;
        double r1418934 = fma(r1418929, r1418932, r1418933);
        double r1418935 = -r1418933;
        double r1418936 = fma(r1418925, r1418925, r1418935);
        double r1418937 = r1418934 + r1418936;
        double r1418938 = sqrt(r1418937);
        double r1418939 = r1418938 - r1418925;
        double r1418940 = r1418939 / r1418928;
        double r1418941 = 2.0;
        double r1418942 = r1418940 / r1418941;
        double r1418943 = -2.0;
        double r1418944 = r1418931 / r1418925;
        double r1418945 = r1418943 * r1418944;
        double r1418946 = r1418945 / r1418941;
        double r1418947 = r1418927 ? r1418942 : r1418946;
        return r1418947;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 6.961131476357277e-08

    1. Initial program 12.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified12.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied prod-diff12.8

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -a \cdot \left(4 \cdot c\right)\right) + \mathsf{fma}\left(-a, 4 \cdot c, a \cdot \left(4 \cdot c\right)\right)}} - b}{a}}{2}\]

    if 6.961131476357277e-08 < b

    1. Initial program 44.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified44.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}}\]

    3. Taylor expanded around inf 11.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, a \cdot \left(4 \cdot c\right)\right) + \mathsf{fma}\left(b, b, -a \cdot \left(4 \cdot c\right)\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))